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Look at Pardo - The isomorphism problem for Higman–Thompson groups for a definition.The groups are denoted by $G_{m,n}^+$ and were originally defined by Higman in his paper. These are similar to the R.Thompson group $V=G^+_{2,1}$ defined earlier by Thompson and used first by McKenzie and Thompson. Hence the name (which is not very fair to Thompson because he was first). These can be defined as the automorphism groups of free algebras in certain (Leavitt) varieties of algebras (in the Thompson case $(2,1)$ this is the Jónsson–Tarski variety). In the notation $G^+_{m,r}$, $m$ defines the variety, $r$ is the rank of the free algebra, $^+$ denotes the derived subgroup. In the Thompson case the algebras have signature $(\mu,\lambda,\rho)$ where $\mu$ is binary, $\lambda,\rho$ are unary and $\mu(\lambda(x),\rho(x))=x, \lambda(\mu(x,y))=x$, and $\rho(\mu(x,y))=y$. Thus if $A$ is a Jónsson–Tarski algebra, then $\mu$ is a bijection between $A\times A$ and $A$. These are finitely presented simple groups. It turned out that many groups $G^+_{m,n}$ are isomorphic (first proved by Higman), and the paper by Pardo describes all situations when that happens: $G_{m,r}^+\cong G_{n,s}^+$ iff $m=n, \gcd(r,n − 1) = \gcd(s,n − 1)$.

Look at Pardo - The isomorphism problem for Higman–Thompson groups for a definition.The groups are denoted by $G_{m,n}^+$ and were originally defined by Higman in his paper. These are similar to the R.Thompson group $V=G^+_{2,1}$ defined earlier by Thompson and used first by McKenzie and Thompson. Hence the name (which is not very fair to Thompson because he was first). These can be defined as the automorphism groups of free algebras in certain (Leavitt) varieties of algebras (in the Thompson case $(2,1)$ this is the Jónsson–Tarski variety). In the notation $G^+_{m,r}$, $m$ defines the variety, $r$ is the rank of the free algebra, $^+$ denotes the derived subgroup. In the Thompson case the algebras have operations $(\mu,\lambda,\rho)$ where $\mu$ is binary, $\lambda,\rho$ are unary and $\mu(\lambda(x),\rho(x))=x, \lambda(\mu(x,y))=x$, and $\rho(\mu(x,y))=y$ (somewhat similar to Hopf algebras. Thus if $A$ is a Jónsson–Tarski algebra, then $\mu$ is a bijection between $A\times A$ and $A$. These are finitely presented simple groups. It turned out that many groups $G^+_{m,n}$ are isomorphic (first proved by Higman), and the paper by Pardo describes all situations when that happens: $G_{m,r}^+\cong G_{n,s}^+$ iff $m=n, \gcd(r,n − 1) = \gcd(s,n − 1)$.

Look at Pardo - The isomorphism problem for Higman–Thompson groups for a definition.The groups are denoted by $G_{m,n}^+$ and were originally defined by Higman in his paper. These are similar to the R.Thompson group $V=G^+_{2,1}$ defined earlier by Thompson and used first by McKenzie and Thompson. Hence the name (which is not very fair to Thompson because he was first). These can be defined as the automorphism groups of free algebras in certain (Leavitt) varieties of algebras (in the Thompson case $(2,1)$ this is the Jónsson–Tarski variety). In the notation $G^+_{m,r}$, $m$ defines the variety, $r$ is the rank of free algebra, $^+$ denotes the derived subgroup. In the Thompson case the algebras have signature $(\mu,\lambda,\rho)$ where $\mu$ is binary, $\lambda,\rho$ are unary and $\mu(\lambda(x),\rho(x)=x, \lambda(\mu(x,y))=x$, and $\rho(\mu(x,y))=y$. These are finitely presented simple groups. It turned out that many groups $G^+_{m,n}$ are isomorphic (first proved by Higman), and the paper by Pardo describes all situations when that happens: $G_{m,r}^+\cong G_{n,s}^+$ iff $m=n, \gcd(r,n − 1) = \gcd(s,n − 1)$.

Look at Pardo - The isomorphism problem for Higman–Thompson groups for a definition.The groups are denoted by $G_{m,n}^+$ and were originally defined by Higman in his paper. These are similar to the R.Thompson group $V=G^+_{2,1}$ defined earlier by Thompson and used first by McKenzie and Thompson. Hence the name (which is not very fair to Thompson because he was first). These can be defined as the automorphism groups of free algebras in certain (Leavitt) varieties of algebras (in the Thompson case $(2,1)$ this is the Jónsson–Tarski variety). In the notation $G^+_{m,r}$, $m$ defines the variety, $r$ is the rank of the free algebra, $^+$ denotes the derived subgroup. In the Thompson case the algebras have signature $(\mu,\lambda,\rho)$ where $\mu$ is binary, $\lambda,\rho$ are unary and $\mu(\lambda(x),\rho(x))=x, \lambda(\mu(x,y))=x$, and $\rho(\mu(x,y))=y$. Thus if $A$ is a Jónsson–Tarski algebra, then $\mu$ is a bijection between $A\times A$ and $A$. These are finitely presented simple groups. It turned out that many groups $G^+_{m,n}$ are isomorphic (first proved by Higman), and the paper by Pardo describes all situations when that happens: $G_{m,r}^+\cong G_{n,s}^+$ iff $m=n, \gcd(r,n − 1) = \gcd(s,n − 1)$.

Look at Pardo - The isomorphism problem for Higman–Thompson groups for a definition.The groups are denoted by $G_{m,n}^+$ and were originally defined by Higman in his paper. These are similar to the R.Thompson group $V=G^+_{2,1}$ defined earlier by Thompson and used first by McKenzie and Thompson. Hence the name (which is not very fair to Thompson because he was first). These can be defined as the automorphism groups of free algebras in certain (Leavitt) varieties of algebras (in the Thompson case $(2,1)$ this is the Jónsson–Tarski variety). It turned out that many groups $G_{m,n}$ are isomorphic (first proved by Higman), and the paper by Pardo describes all situations when that happens: $G_{m,r}^+\cong G_{n,s}^+$ iff $m=n, \gcd(r,n − 1) = \gcd(s,n − 1)$.

Look at Pardo - The isomorphism problem for Higman–Thompson groups for a definition.The groups are denoted by $G_{m,n}^+$ and were originally defined by Higman in his paper. These are similar to the R.Thompson group $V=G^+_{2,1}$ defined earlier by Thompson and used first by McKenzie and Thompson. Hence the name (which is not very fair to Thompson because he was first). These can be defined as the automorphism groups of free algebras in certain (Leavitt) varieties of algebras (in the Thompson case $(2,1)$ this is the Jónsson–Tarski variety). In the notation $G^+_{m,r}$, $m$ defines the variety, $r$ is the rank of free algebra, $^+$ denotes the derived subgroup. In the Thompson case the algebras have signature $(\mu,\lambda,\rho)$ where $\mu$ is binary, $\lambda,\rho$ are unary and $\mu(\lambda(x),\rho(x)=x, \lambda(\mu(x,y))=x$, and $\rho(\mu(x,y))=y$. These are finitely presented simple groups. It turned out that many groups $G^+_{m,n}$ are isomorphic (first proved by Higman), and the paper by Pardo describes all situations when that happens: $G_{m,r}^+\cong G_{n,s}^+$ iff $m=n, \gcd(r,n − 1) = \gcd(s,n − 1)$.